View source for Real algebraic variety

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800201.png" /> of real points of an [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800202.png" /> defined over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800203.png" /> of real numbers. A real algebraic variety is said to be non-singular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800204.png" /> is non-singular. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800205.png" /> is a smooth variety, and its dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800206.png" /> is equal to the dimension of the complex variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800207.png" />; the latter is known as the complexification of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800208.png" />.

Non-singular regular complete intersections have been most thoroughly studied. These are varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r0800209.png" /> in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002010.png" /> which are non-singular regular intersections of hypersurfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002013.png" /> is a homogeneous real polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002014.png" /> variables of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002015.png" />. In such a case the matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002016.png" /></td> </tr></table>

has rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002017.png" /> at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002019.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002020.png" /> denote the real algebraic variety defined as the intersection system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002021.png" /></td> </tr></table>

Examples of regular complete intersections are:

1) A plane real algebraic curve; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002025.png" />.

2) A real algebraic hypersurface; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002028.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002029.png" />, a real algebraic surface is obtained.

3) A real algebraic space curve; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002031.png" />. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002032.png" /> is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002033.png" />, while the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002034.png" /> is cut out on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002035.png" /> by a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002036.png" />.

A real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002037.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002038.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002039.png" /> consists of finitely many components diffeomorphic to a circle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002040.png" /> is even, these components are all two-sidedly imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002041.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002042.png" /> is odd, one component is imbedded one-sidedly, while the remaining ones are imbedded two-sidedly. A two-sidedly imbedded component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002043.png" /> is called an oval of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002044.png" />. An oval lying inside an odd number of other ovals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002045.png" /> is called odd, while the remaining ovals are even.

The number of components of a plane real algebraic curve of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002046.png" /> is not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002047.png" /> (Harnack's theorem) [[#References|[1]]]. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002048.png" /> there exists a plane real algebraic curve with this largest number of components — the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002050.png" />-curve. (For methods of constructing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002051.png" />-curves see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]; for a generalization of these results to include space curves, see [[#References|[2]]].)

D. Hilbert posed in 1900 the problem of studying the topology of real algebraic varieties, and also of the imbedding of real algebraic varieties into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002052.png" /> and of one real algebraic variety into another (Hilbert's 16th problem). He also pointed out difficult partial problems: the study of the mutual locations of the ovals of a sixth-order curve, and the topology and the imbedding of a real algebraic fourth-order surface into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002053.png" />. These partial problems have been solved [[#References|[12]]], [[#References|[13]]].

For a plane real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002054.png" /> of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002055.png" /> the following exact inequality is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002056.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002057.png" /> is the number of even ovals and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002058.png" /> is the number of odd ovals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002059.png" /> (Petrovskii's theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002060.png" /> is odd, a similar inequality is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002062.png" /> is a straight line in general position [[#References|[4]]]. When these results are generalized to include the case of a real algebraic hypersurface of even order, the role of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002063.png" /> is played by the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002065.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002066.png" /> is odd, the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002067.png" /> is played by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002068.png" />. Thus, for a real algebraic hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002069.png" /> of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002070.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002071.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002072.png" /> is the number of terms of the polynomial

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002073.png" /></td> </tr></table>

of degree not higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002074.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002075.png" /> is odd, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002076.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002077.png" /></td> </tr></table>

[[#References|[5]]]. The following inequality is satisfied for a real algebraic space curve (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002078.png" />) for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002079.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002080.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002081.png" /></td> </tr></table>

(if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002082.png" />, this estimate is exact [[#References|[6]]]). Petrovskii's theorem has been generalized to arbitrary real algebraic varieties .

For a plane real algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002083.png" />-curve of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002084.png" /> the following congruence is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002085.png" /></td> </tr></table>

[[#References|[8]]], , [[#References|[13]]]. In proving this congruence ([[#References|[8]]], ), real algebraic varieties were studied by methods of differential topology in a form which opened the way for further investigations. Let the plane real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002086.png" /> have even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002087.png" /> and let the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002088.png" /> be chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002089.png" /> is orientable, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002090.png" /> denote, respectively, the number of ovals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002091.png" /> which externally bound the components of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002092.png" /> with positive, zero and negative Euler characteristics. In a similar manner, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002093.png" /> are the numbers of such odd ovals for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002094.png" />. Then [[#References|[8]]], [[#References|[3]]],

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002095.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002096.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002097.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002098.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r08002099.png" /></td> </tr></table>

For an arbitrary real algebraic variety in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020100.png" />-dimensional projective space the following inequality is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020102.png" /> is the homology space of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020103.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020104.png" /> . This inequality is a generalization of Harnack's theorem. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020105.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020106.png" /> is always an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020107.png" /> is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020108.png" />-variety. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020110.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020111.png" />-variety.

The validity of the following congruences has been demonstrated:

A) For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020112.png" />-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020113.png" /> and for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020114.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020115.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020116.png" /> is the [[Signature|signature]] of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020117.png" /> .

B) For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020118.png" />-variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020119.png" /> and even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020120.png" /> ([[#References|[13]]]):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020121.png" /></td> </tr></table>

cf. the overview [[#References|[3]]].

C) For a regular complete intersection, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020122.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020123.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020124.png" />-variety and the inclusion homomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020125.png" /></td> </tr></table>

is zero, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020126.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020127.png" /></td> </tr></table>

In this case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020128.png" /> is even, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020129.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020130.png" />-variety and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020131.png" /> is zero ([[#References|[11]]]):

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020132.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020134.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020135.png" />,

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020136.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020138.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020139.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020140.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020141.png" />,

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020142.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020144.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020145.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020146.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020147.png" />.

In particular, for a real algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020148.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020149.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020150.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020151.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020152.png" />-surface, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020153.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020154.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020155.png" />-surface, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020156.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020157.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020158.png" />-surface and contracts to a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020159.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020160.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020161.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020162.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020163.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020164.png" />-surface and contracts to a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020165.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020166.png" /></td> </tr></table>

Certain congruences have also been proved , [[#References|[13]]] for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020167.png" />. In particular, for a plane real algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020168.png" /> which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020169.png" />-curve of even order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020170.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020171.png" /></td> </tr></table>

Certain results have also been obtained [[#References|[13]]] for real algebraic varieties with singularities. For an interesting approach to the study of real algebraic varieties see [[#References|[14]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Harnack, "Ueber die Vieltheitigkeit der ebenen algebraischen Kurven" ''Math. Ann.'' , '''10''' (1876) pp. 189–198</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Hilbert, "Ueber die reellen Züge algebraischer Kurven" ''Math. Ann.'' , '''38''' (1891) pp. 115–138</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Hilbert, "Mathematische Probleme" ''Arch. Math. Phys.'' , '''1''' (1901) pp. 213–237 {{MR|}} {{ZBL|32.0084.05}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.G. Petrovskii, "On the topology of real plane algebraic curves" ''Ann. of Math.'' , '''39''' : 1 (1938) pp. 189–209 {{MR|1503398}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.A. Oleinik, I.G. Petrovskii, "On the topology of real algebraic surfaces" ''Transl. Amer. Math. Soc.'' , '''7''' (1952) pp. 399–417 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''13''' (1949) pp. 389–402 {{MR|0048095}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> O.A. Oleinik, "On the topology of real algebraic curves on an algebraic surface" ''Mat. Sb.'' , '''29''' (1951) pp. 133–156 (In Russian) {{MR|44863}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> , ''Hilbert problems'' , Moscow (1969) (In Russian) {{MR|}} {{ZBL|0187.35502}} {{ZBL|0186.18601}} {{ZBL|0181.15503}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.I. Arnol'd, "Distribution of the ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms" ''Funct. Anal. Appl.'' , '''5''' : 3 (1971) pp. 169–176 ''Funkts. Anal.'' , '''5''' : 3 (1971) pp. 1–9 {{MR|}} {{ZBL|0268.53001}} </TD></TR><TR><TD valign="top">[9a]</TD> <TD valign="top"> V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" ''Funct. Anal. Appl.'' , '''6''' : 4 (1972) pp. 301–306 ''Funkts. Anal.'' , '''6''' : 4 (1972) pp. 58–64</TD></TR><TR><TD valign="top">[9b]</TD> <TD valign="top"> V.A. Rokhlin, "Congruences modulo 16 in Hilbert's sixteenth problem" ''Funct. Anal. Appl.'' , '''7''' : 2 (1973) pp. 163–165 ''Funkts. Anal.'' , '''7''' : 2 (1973) pp. 91–92</TD></TR><TR><TD valign="top">[10a]</TD> <TD valign="top"> V.M. Kharlamov, "A generalized Petrovskii inequality" ''Funct. Anal. Appl.'' , '''8''' : 2 (1974) pp. 132–137 ''Funkts. Anal.'' , '''8''' : 2 (1974) pp. 50–56 {{MR|}} {{ZBL|0301.14021}} </TD></TR><TR><TD valign="top">[10b]</TD> <TD valign="top"> V.M. Kharlamov, "A generalized Petrovskii inequality II" ''Funct. Anal. Appl.'' , '''9''' : 3 (1975) pp. 266–268 ''Funkts. Anal.'' , '''9''' : 3 (1975) pp. 93–94</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.M. Kharlamov, "Additive congruences for the Euler characteristic of real algebraic manifolds of even dimensions" ''Funct. Anal. Appl.'' , '''9''' : 2 (1975) pp. 134–141 ''Funkts. Anal.'' , '''9''' : 2 (1975) pp. 51–60</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.M. Kharlamov, "The topological type of nonsingular surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080020/r080020172.png" /> of degree four" ''Funct. Anal. Appl.'' , '''10''' : 4 (1976) pp. 295–304 ''Funkts. Anal.'' , '''10''' : 4 (1976) pp. 55–68 {{MR|}} {{ZBL|0362.14013}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> D.A. Gudkov, "The topology of real projective algebraic varieties" ''Russian Math. Surveys'' , '''29''' : 4 (1974) pp. 1–80 ''Uspekhi Mat. Nauk'' , '''29''' : 4 (1974) pp. 3–79 {{MR|0399085}} {{ZBL|0316.14018}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> D. Sullivan, "Geometric topology" , '''I. Localization, periodicity, and Galois symmetry''' , M.I.T. (1971) {{MR|0494074}} {{MR|0494075}} {{ZBL|1078.55001}} {{ZBL|0871.57021}} {{ZBL|0366.57003}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Viro, "Successes of the last five years in the topology of real algebraic varieties" , ''Proc. Internat. Congress Mathematicians (Warszawa, 1983)'' , PWN &amp; North-Holland (1984) pp. 603–619</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Wilson, "Hilbert's sixteenth problem" ''Topology'' , '''17''' (1978) pp. 53–74</TD></TR></table>